Question

question let $h(x)=f(g(x))$. if $g(3)= - 4,g(3)=5,f(-4)= - 6,$ and $f(-4)=4$, find $h(3)$. do not include \$h(3)=$\ in your answer. provide your answer below:

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step2: Substitute $x = 3$

When $x = 3$, we have $h^{\prime}(3)=f^{\prime}(g(3))\cdot g^{\prime}(3)$.

Step3: Use given values

We know that $g(3)=-4$ and $g^{\prime}(3)=5$, and $f^{\prime}(-4) = 4$. Substituting $g(3)=-4$ into $f^{\prime}(g(3))$ gives $f^{\prime}(-4)$. So $h^{\prime}(3)=f^{\prime}(-4)\cdot g^{\prime}(3)$.

Step4: Calculate the result

$h^{\prime}(3)=4\times5 = 20$.

Answer:

20